Journal of Symbolic Logic

An incomplete set of shortest descriptions

Frank Stephan and Jason Teutsch

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The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree by identifying an acceptable set of domain-random strings within each degree.

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J. Symbolic Logic, Volume 77, Issue 1 (2012), 291-307.

First available in Project Euclid: 20 January 2012

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Stephan, Frank; Teutsch, Jason. An incomplete set of shortest descriptions. J. Symbolic Logic 77 (2012), no. 1, 291--307. doi:10.2178/jsl/1327068704.

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  • Manuel Blum On the size of machines, Information and Control, vol. 11(1967), pp. 257–265.
  • Cristian S. Calude and André Nies Chaitin $\Omega$ numbers and strong reducibilities, Journal of Universal Computer Science, vol. 3(1997), no. 11, pp. 1162–1166 (electronic).
  • John Case Homework assignment for students, Computer and Information Sciences Department, University of Delaware,1990.
  • Gregory J. Chaitin A theory of program size formally identical to information theory, Journal of the ACM, vol. 22(1975), pp. 329–340.
  • –––– Incompleteness theorems for random reals, Advances in Applied Mathematics, vol. 8(1987), no. 2, pp. 119–146.
  • George Davie Foundations of mathematics–-recursion theory question, family pipermail/fom/2002-May/005535.html.
  • Rodney G. Downey and Denis R. Hirschfeldt Algorithmic randomness and complexity, Theory and Applications of Computability, Springer,2010.
  • Stephen Fenner and Marcus Schaefer Bounded immunity and btt-reductions, Mathematical Logic Quarterly, vol. 45(1999), no. 1, pp. 3–21.
  • Santiago Figueira, Frank Stephan, and Guohua Wu Randomness and universal machines, Journal of Complexity, vol. 22(2006), pp. 738–751.
  • Harvey Friedman Foundations of mathematics–-recursion theory question, family
  • Edward R. Griffor Handbook of computability theory, Studies in Logic and Foundations of Mathematics, North-Holland, Amsterdam,1999.
  • Sanjay Jain, Frank Stephan, and Jason Teutsch Index sets and universal numberings, CiE 2009: Proceedings of the 5th conference on Computability in Europe, Springer,2009, pp. 270–279.
  • Efim Kinber On btt-degrees of sets of minimal numbers in Gödel numberings, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23(1977), no. 3, pp. 201–212.
  • Martin Kummer personal communication.
  • –––– On the complexity of random strings $($extended abstract$)$, STACS '96: Proceedings of the 13th annual Symposium on Theoretical Aspects of Computer Science, Springer,1996, pp. 25–36.
  • Martin Kummer and Frank Stephan Recursion-theoretic properties of frequency computation and bounded queries, Information and Computation, vol. 120(1995), no. 1, pp. 59–77.
  • G. B. Marandžjan On the sets of minimal indices of partial recursive functions, Mathematical Foundations of Computer Science, vol. 74(1979), pp. 372–374.
  • –––– Selected topics in recursive function theory, Technical Report 1990-75, Technical University of Denmark, August1990.
  • Per Martin-Löf The definition of random sequences, Information and Control, vol. 9(1966), pp. 602–619.
  • Albert R. Meyer Program size in restricted programming languages, Information and Control, vol. 21(1972), pp. 382–394.
  • Piergiorgio Odifreddi Classical recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland, Amsterdam,1989.
  • Robert W. Robinson Interpolation and embedding in the recursively enumerable degrees, Annals of Mathematics (2), vol. 93(1971), pp. 285–314.
  • Gerald E. Sacks Recursive enumerability and the jump operator, Transactions of the Amererican Mathematical Society, vol. 108(1963), pp. 223–239.
  • –––– The recursively enumerable degrees are dense, Annals of Mathematics (2), vol. 80(1964), pp. 300–312.
  • Marcus Schaefer A guided tour of minimal indices and shortest descriptions, Archive for Mathematical Logic, vol. 37(1998), no. 8, pp. 521–548.
  • Steven Schwarz Quotient lattices, index sets, and recursive linear orderings, Ph.D. thesis, University of Chicago,1982.
  • Robert I. Soare Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer,1987.
  • Frank Stephan Degrees of computing and learning, Habilitationsschrift, University of Heidelberg,1999.
  • –––– The complexity of the set of nonrandom numbers, Randomness and complexity, from Leibnitz to Chaitin (Cristian S. Calude, editor), World Scientific,2007, pp. 217–230.
  • Frank Stephan and Jason Teutsch Immunity and hyperimmunity for sets of minimal indices, Notre Dame Journal of Formal Logic, vol. 49(2008), no. 2, pp. 107–125.
  • Jason Teutsch Noncomputable spectral sets, Ph.D. thesis, Indiana University,2007.
  • –––– On the Turing degrees of minimal index sets, Annals of Pure Applied Logic, vol. 148(2007), no. 1–3, pp. 63–80.
  • A. K. Zvonkin and L. A. Levin The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, Russian Mathematical Surveys, vol. 25(1970), no. 6, pp. 83–124.